Download Cell Mechanics

Cell Mechanics
Prof. M.S. Ju
Dept. of mechanical Engineering
National Cheng Kung University
Tainan
Outline
I. Introduction & biological basis for cell mechanics (1)
II. Experimental measurements of intracellular
mechanics(2)
III. Continuum elastic or viscoelastic models for the cell(4)
IV. Models of cytoskeletal mechanics based on tensegrity
(6)
V. Active cellular protrusion: continuum theories and
models (10)
From: Mofrad & Kamm’s
1. Introduction


Cells - basic functional units of life; numerous
components with distinct mechanical
characteristics (Prof. Tseng’s)
Cell mechanics
Macroscopic events: cell shape, motility, adhesion,
deformation
 Microscopic events: sense mechanical signals &
transduce them to a cascade of biochemical signals &
leading to biological response



Goals

Describe and evaluate mechanical properties of cell and
cellular structure

Mechanical interaction between cell & their environment
Recent focus

Rheology of cytoskeleton and reconstituted gels of major
cytoskeletal compopnents: actin filaments, intermediate
filaments, microtubules, and their cross-linking proteins

Mechanical interaction of cell with surroundings and how
the interaction causes changes in morphology and
biological signal that lead to functional adaptation or
pathological condition


Computational models

Finite element – based continuum models for cell
deformation

Actin filament-based model for cell motility
Experimental techniques


Mechanical perturbation of cell by imposing
deformation/force vs static/dynamic response
Theories for mechanics of living cells

simple elastic, viscoelastic or poro-viscoelastic continuum

Porous gel or soft glassy material

Tensegrity (tension integrity) network incorporating
discrete structural elements (compression bearing)

Test condition, length scale
Objectives



Providing background information on
environment surrounding a cell
Molecular building blocks used to impart
structural strength to the cell
Importance of cell mechanics in biological
function (mechanobiology)
1-2 Role of cell mechanics in
biological function




Focused primary on eukaryotic cells of animals
Exception – red blood cell or erythrocyte which
do not have nucleus
Living cells have time-varying properties
Changing entities with capability to alter
mechanical properties in response to external
stimuli.
Maintenance of cell shape

Function of cell depends on its shape

Shape is maintained through structural stiffness

Examples:

biconcave erythrocyte,

spherical leucocytes – roll along vascular endothelium before
adhere and migrate into tissue

Neuronal cells extend long processes to conduct signal

Airway epithelial cells covered with a bed of cilia, finger-like
cell extensions; propel mucus along airway of lung
Internal structure
along with cell
membrane provide
structural integrity to
maintain needed
shape to accomplish
its function
Cell Migration
Cell migrates during :
 Development – organism grows its various parts
 Wound repair – cells from undamaged tissue
migrate into the wound & renew tissue
 Combating infection – cells of immune system
transmigrate from vascular system across vessel
wall into infected tissue
Stages of cell migration




Protrusion: extension of cell at
leading edge in movement
direction
Adhesion of protrusion to
surrounding substrate or matrix
Contraction of cell that transmits
a force from protrusions at leading
edge to cell body, pull it forward
Release of attachment at the rear
Note: cell senses biochemical cues- gradient of chemotactic agents and
physical environment, e.g. variations in stiffness of substrate
Passive stiffness & active contractility
Mechanosensing


Many cells are able to sense & respond to
externally applied forces
The responses
changes in membrane channel activity
 up- or down – regulation of gene expression
 alterations in protein synthesis
 altered cell morphology


Example: hair cells of inner ear
Hair cells


Stereocilia extend from
apical surface to form
bundles
As stereocilia bundles
move in response to
fluid oscillations in
cochlea, tension in tip
link increases, opening an
ion channel to initiate
electrochemical response.
Ca++
Other mechanisms



Conformational changes in intracellular proteins due to
transmission of external forces to cell interior, leading
to changes in reaction rates through a change in binding
affinity
Changes in viscosity of cell membrane, altering rate of
diffusion of trans-membrane proteins & their rates
Direct transmission of force to nucleus and to
chromatin contained inside, affecting expression of
specific genes.
Force experienced by endothelial
Glycocalyx: 醣外被
Stress responses and role of
mechanical forces in disease




Physical forces are instrumental in remodeling
of tissues
Bone remodeling/modeling, Wolff’s law
Cells can sense and respond to mechanical
stimulus
Atherosclerosis, arthritis and pulmonary
hypertension
Active Cell Contraction



Vascular smooth muscle cells, cardiac myocytes &
skeletal muscle cells can generate force
Common force generation mechanism: molecular
motor comprised of actin and myosin (sacomere)
Other cells contains contractile machinery for
maintaining resting level of cell tension, changing
cell shape or cell migration.
Cardiac myocytes in culture
striation
Structural anatomy of a cell



Cells are biologically active & their structure
reflects or responds to physical environment.
(primary distinction to inert materials)
Thermal fluctuations need to be considered.
Affects biochemical processes that lead to
Intracellular remodeling
 Elastic characteristics of membrane
 Filaments of cytoskeleton




Cells do not constitute structural elements of
tissue
Mechanical stiffness of resident cells has less
contribution to modulus of tissue
Deformation dictated by surrounding matrix
Collagen & hydroxyapatite in bone
 Collagen & protoglycan w. high negative charge in
cartilage



In muscle, contractile force and modulus of tissue
are dominated by cellular activity.
In arterial wall or pulmonary airways, collagen and
elastin filaments in extracellular matrix balance
bulk of stress.

In cardiac tissue, myocytes constitute large
fraction of tissue volume and responsible for
stresses & deformations of myocardium that are
time-varying
Extracellular matrix & its attachment
to cells


Force in vivo transmitted to cell via extracellular matrix
(ECM) which shares load-supporting function.
Cell membrane receptors’ extracellular domains bind to
various proteins of ECM.



Integrins to fibronectin, vitronectin, collagen and laminin
Intracellular domains bind to cytoskeleton directly or
indirectly.
Other adhesion molecules bind to ECM, base
membrane, neighboring cells or cells suspended in
flowing blood.
Adhesion molecules
Transmission of force to cytoskeleton and role
of lipid bilayer



Cell membrane – thin lipid bilayer consisted of
mix of phospholipids, glycolipids, cholesterol
and transmembrane proteins (50% w.)
Phospholipids are amphipathic
Integral membrane proteins – ion channels,
pathway for transmembrane signaling, structural
bridge- direct adhesion between cytoskeleton &
extracellular matrix.




Membrane – barrier isolating cell interior from
extracellular environment & maintaining
biochemical condition.
Bilayer contribute less to overall stiffness of cell.
Fluid mosaic model (Singer & Nicolson 1972): 2-D
fluid within which integral membrane proteins diffuse.
Thickness ~ 6 nm, area modulus ~ 0.1-1.0 N/m
(lipid bilayer) 0.45N/m (RBC), rupture strength
0.01-0.02 N/m for RBC & lipid bilayer


Membrane bending rigidity: 2-4x10-19 Nm for
RBC, 1-2x10-18 for neutrophils
Shear modulus: 10-6 Ns/m for RBC, negligible
for pure lipid bilayers
Intracellular Structures



Intracellular structures influence material
properties of cell: cortex, projections, internal
structure (nucleus, actin filaments, microtubules,
intermediate filaments)
Cortex: dense structure adjacent to membrane,
Examples
RBC, filamentous protein, spectrin, shape rigidity
 Epithelial cells, microvili in intestine; cilia in lung



Nucleus: not well-known mechanical properties;
separate contribution of nuclear envelope, and
nucleoplasm.
Migrating cells:
leading edge sends out protrusions (lamellipodia or
filopodia) rich in actin & highly cross-linked.
 Dynamics of actin polymerization & depolymerization
 Active contraction of network due to actin-myosin
interactions

*Dr. Lin’s PPT
Actin filaments





Form by polymerization of globular, monomeric actin
(G-actin) into twisted strand of filamentous actin (Factin) 7-9 nm f with barbed end & pointed end
ATP can bind to barbed end allows for monomer
addition & filament growth
Depolymerization occurs at pointed end
Filament growth regulated by ionic concentrations,
capping, binding, branching, and severing proteins.
ATP (adenosine triphosphate)腺膘呤核苷三磷酸

From actin filaments two structures
Stress fiber: tertiary structures, fiber bundle
 3D lattice-like network can be formed throughactinbinding proteins (ABP)




APB Examples: fimbrin & a-actinin, connect
filaments into a 3D space filling matrix (gel) with
filaments jointed nearly at right angle.
Actin constitutes from 1-10% of all proteins in
most cells, higher in muscle cells (primary
structural component)
Table 1-2 elastic properties
Microtubules





Second major constitutent of cytoskeleton.
Polymerized filaments constructed from
monomers of a- & b-tubulin in helical
arrangement (55 kDa)
Hollow cylinder, OD 25nm, high bending
stiffness, persistent length 6mm, Young’s
modulus similar to actin
Long slender structures: cilia & flagella
Network for transporting chromosomes during
cell division


Microtubules are highly dynamic, constant
polymerization & depolymerization, half-life
only few minutes
Asymmetric growth,
Intermediate Filaments

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

A family of proteins > 50 members
Common structure: central a helical domain of
>300 residues form coil; dimers assembled into
tetramers; forming protofilaments; bundle into
rope-like structure, 8 protofilaments with
persistent length = 1 um
Long-term stability & high resistance to
solubility in salts
Form without GTP or ATP hydrolysis


Intermediate filaments labeled with fluorescent
marker can be used as fiducial markers for study
the strain field within the cells
High concentration in entire cell
Linking Proteins

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


Actin, microtubules & intermediate filaments are
associated with cytoskeleton
Linking proteins within cytoskeleton affect
strength & integrity of resulting matrix.
Cell membrane, nuclear membrane & all
organelles & other intracellular bodies also affect
mechanical response of a cell.
Complexity of intracellular structure
Example: focal adhesion complex



Forces transmitted from extracellular matrix
(fibronectin) via integral membrane adhesion receptors
(a- and b-integrins), membrane associated proteins,
actin-binding proteins to cytoskeleton
Active contraction is fundamental feature of
cytoskeleton; muscle and most cells contain contractile
machinery
Force measured in resting fibroblasts, stress in focal
adhesion 5 Kpa!
1-3 Overview




Full spectrum of views on current approaches to
modeling cell mechanics.
Diversity of background: biophysics, bioengineering
and physical chemistry.
Diversity of approaches: finite-element methods
(continuum mechanics), cross-linked polymer network
(cytoskeleton), soft glassy materials & gels.
Static, instantaneous nature of structure and dynamic
nature due to polymerization and biological processes.
II. Experimental Measurements of
Intracellular Mechanics

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Novel methods to viscoelasticity of soft materials and new
theories; apply pN forces & measure nm displacement
Methods: optical trap, magnetic beads, glass needle, atomic
force microscope
Thermal motion of refractive particles, vescicles,
submicron-beads
Challenge: relate experimental and theoretical results
derived from molecular scale to measurement on
macroscopic scale
Discrepancy between nano-scale rheology & bulk
measurement
2-1 Introduction

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Cell mechanical processes: locomotion,
secretion, cell division
Plant cells & bacteria: hard cell wall
Animal cells: soft membrane & cytoskeleton
(internal protein network)
Cytoplasm: glassy, bio-solid, both viscous and
elastic characteristics
Measurement methods
Forces to which cells are exposed in
biological context

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

Large range of stress to different tissues
Cytoskeleton structures: response passively to
provide strength and sensing external forces &
their cellular responses
Factors: molecular sensors, signaling pathways,
rheological parameters
Cartilage stress ~ 20MPa, chondrocytes regulate
glycosaminoglycans(GAG) & others
Bone & osteocytes respond to similar level;

Endothelial cells ~ shear stress < 1 Pa, wide range of
morphological & transcriptional changes

Neutrophils activate in response to smaller shear stress

Magnitude, geometry & time course of mechanical
perturbations are critical

Tendons, skeletal muscles experience or generate uni-axial
force & deformation

Blood vessel lining cells experienced shear stress due to
flow; respond to changes in stress rather than magnitude

Above cells & epithelial cells in lung experience large-areadilation forces, both magnitude & temporal characteristics
are critical
2-2 Macrorheology, diffusion &
sedimentation methods



Challenges: small size of cell, heterogeneous
structure of cell interior, active remodeling of
cytoplasm (constitutive changes, response to
external force)
Stronger mechanical stimulus yield more
biochemical reaction
Small strain(<10%) for linear response and large
strain for nonlinear response
Whole cell aggregates

Employ standard rheology instruments to measure
stress/strain relationship on macroscopic sample
containing many cells (single type)

Muscle fibers: actin/myosin based fibers arranged in
parallel & attached longitudinally

Excellent agreement between single molecule
measurement of force-elongation relation with
macroscopic compliance measurement of muscle fibers

sedimented samples of a single cell type, assumption:
deformation is related to cell interior rather than sliding
of cells past each other.

Melanoma (黑色瘤(皮膚癌))cell, dictyostelium cell:
single actin-binding protein mutations

Disadvantages: need to verify how cells attach to
each other & contribution of cell membrane or
extracellular matrix.
Sedimentation by particles



To explore variations of viscoelastic properties
within a cell, probes with size smaller than inhomogeneity are used.
Diffusion or sedimentation of intracellular
granules with high specific gravity in cytoplasm
are observed.
Large cells containing colored or retractile
particles. (1950s)




Record rate of falling of starch grains within a bean cell
& compare with same particle in fluids of known
densities & viscosities.
Note viscosity of fluid can be measured by the
viscometers introduced in Chapter 1.
8 mPa s for cytoplasmic viscosity
Similar to falling-sphere method in macroscopic
rheometry (Rockwell et al. 1984)
2 g ( -  ) r
v
9
2
v: velocity of particle
r: particle radius
g: gravity constant
: viscosity of cytoplasm
: density of particle
: density of cytoplasm



By injecting a small droplet of inert oil into a
large cell (muscle fiber) and observe rising rate
of the droplet, 29 mPa.s
Organelles can be made to sediment by
gravitational force in a centrifuge
Oocyte, amoebas, slime molds have cytoplasmic
viscosities 2 ~ 20 mPa.s
Diffusion



First centrifuging a large cell such as sea urchin egg or
amoeba then monitor displacement of a single particle
of radius r in one direction x(t); 4 mPa.s (cP) four times
of water
Sedimentation force 100~5000 times of gravity;
sufficient for intracellular organelles to get concentrated
while cell remains intact.
Stokes-Einstein relation:
kB T
x (t ) 
3  r
2
< > ensemble average
kB Boltzmann constant
T absolute temperature

Three important features of intracellular
material properties:
Apparent viscosity of protoplasm depended strongly
on flow rates
 Viscous flow of internal organelles could be
measured only deeper inside the cell
 Cellular viscosity is temperature dependent

2-3 Mechanical Indentation of Cell
Surface



Glass micro-needles
Cell poker
Atomic Force Microscopy
Glass micro-needle



To apply force large enough to deform the cell but no
damage of cell
Pull on individual cultured neurons, point forces for
initiating neurite extension
Calibration of clamped wire needle by hanging a weight at
free end and the deflection y
F L3
y ( L) 
,
3 EI
I

 r4
4
The wire needle is used to calibrate a thinner glass and
repeated till a glass microneedle that can detect nN or
smaller force
Cell poker

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
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Cell suspended in fluid
Vertical glass needle
attached to wire needle
that coupled to a
piezoelectric actuator
Vertical displacements of
both end of wire are
measured optically
x = difference of
displacement
F= k x




Resolutions: displacement < 100nm, force < 10 nN
Typical force vs. displacement curve
Elasticity and plastic (?) deformation of cell
Tip is smaller than cell, local viscoelasticity can be
probed at different regions of cell
findings



Large difference in relative stiffness over
different areas
High degree of softening when actin-filamentdisorganization drugs like cytochalasin were
applied
Apparent stiffness increased as amplitude of
indentation increased
Discussion



Hertz model is not valid if cell thickness is not
much greater than indentation depth.
Cell cytoskeleton is not isotropic homogeneous
material
Force exerted on a cell initiate biochemical and
other active reaction (Daily et al. 1984)
Atomic Force Microscopy

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

Imaging by scanning a sharp microscopy tip
over a surface while recording tip deflection
simultaneously (Scanning Probe Microscope)
Deflection time course - surface profile
Three modes: contact mode, tapping mode,
jumping mode & others
AFM indentation can be used to probe
mechanical properties at selected locations
Better spatial & force resolution than cell poking
Atomic Force Microscope System (Agilent 5500 )
57
Contact mode
z

位移感測
A  B  - C  D 

d vertical 
 A  B  C  D
A  C  -  B  D

dlaterial 
 A  B  C  D

回饋系統
接觸模式
User defined Servo and
scanning parameters
y
o
Error
User defined
Set-point
deflection
Controller
Electronics
Laser
Feedback
signal for
PZT
scanner
+
-
Instantaneous
Deflection signal
AB
CD
PZT Scanner for
XYZ movement
Cantilever and tip
Sample
laser
X
Y
u
v
Power
Amplifier
z
Z
PZT
contact
indented
q
A
A
d
Sample
Substrate
F  kd
d  z - d  cos q
d
z
A’
q
d
Indentation of semi-space elastic
solid (Hertz model)

Sphere indenter
Fs 

(2.4)
Conical indenter
Fc 

3
4 E
R
d
3 (1 - 2 )

E
tana  d 2
2
2 (1 - )
(2.5)
Regular Pyramid (Bilodeau 1992)
tan a  E d 2
F  0.7453
1 - 2


Soma of PC-12 cell
topography
deflection
Semi-contact mode (tapping mode)
PZT-xyz
Excitation
source
u(t)
未受力
k
b
Sample
ξ=0
+ξ
Fext(ξ)
A
Ao
m
Liquid
受外力
t
t
-ξ0
• 作用小
6
Amplitude (nm)
0
10
15
15
20
0
0
10
m
15
20
10
20
m
200 400 600 800
20
0
10
m
-5
10
m
20
20
0
(B)
m
200 400 600
800
Topography
(nm)
-5
0(nm) 5
Amplitude
0
A
15
0
10
m
20
15
0
10
m
20
0
20
10
20
m
0
10
20
m
10
12
16
18
20
16
18
E (kPa)
20
10
18
m
0
20
20
Phase delay (degree)
0
10
5
5
15
m
10
20
10
20
15
B
16
5
5
15
m
m
10
20
往左掃描
5
15
20
往左掃描：16.52±0.50 kPa 誤差為1.27%
往右掃描：16.31±0.53 kPa
5
15
E (kPa)
0
10
6
8 (degree)
10
12
Phase
delay
0
0
m
0
10
85
10
12
14
Amplitude (nm)
Vertical
position
(m)
Line
A-B (μm)
0
10
m
m
2
4
6
Topography (nm)
5
20
10
E (kPa) m
6
8
20
0 5
20
10
15
0
E 18
(kPa)
20
0
0
5
18
5
0
16
20
B
0
15
16
15.5
10
20
10
m
16
A
0
10
20
20
0
5
2015
0
18
20
m
m
10
m
18
16
10
6
8 (degree)
10
Phase
delay
E (kPa)
0
Scan0 direction:Left
Scan5 direction:Right
5
Phase delay (degree)
10
0
10
5
15
0
8
m
m
m
5
Topography (nm)
5
15
16.5 20
2
-6Amplitude
-4 -2 (nm)
0 2
0
m
Elastic modulus (kPa)
200 400 600
800
Topography
(nm)
0
10
17 10
20
0
16
0
5
17.5
-2
m
(A)
-4
10
20
15
0
10
m
E (kPa)
往右掃描
0
10
5
15
m
0
-6
m
18
200 400 600 800
m
0
m
• 側向力小
10
20
20
0
15
10
m
20
63

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



Limitations: manipulating through accessible surface
of cells can not measure elastic moduli well inside
cell
Sharp tip (50 nm f) and Hertz model to test various
types of cells (1999).
Cells are inhomogeneous, anisotropic, active biosolids.
Cell periphery are crucial for cell motility, too thin
to apply standard Hertz model
Spatial inhomogeneity of cells (stress fibers,
microtubules) may be filtered by large probe
Different mechanical properties between cell center
and cell periphery

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


Quasi-static or low frequency measurements may
induce active cellular responses in addition to
passive material properties.
High frequency tapping mode are also used but
cell dynamically stiffened by the rapidly
oscillating tip.
Polystyrene beads of proper radius to contact cell
(Mahaffy 2004, 2000)
Bead radius & inhomogeneity relationship
Zero-frequency shear modulus: 1~2 kPa for
fibroblast




Complex modulus at 50-300Hz were measured
and evaluated with extended Hertz model.
Hydrodynamic drag on rest of cantilever is not
easy to compensate.
Hertz model has been modified to account for
finite sample thickness and boundary condition
on substrate.
Applied to thin lamellipodia of cells: Young’s
modulus of 1~2 kPa for fibroblasts
2.4 Mechanical Tension applied to
cell membrane




Pulling of cell membrane by controlled suction within
micropipette to measure viscosity & elastic response of
cells (1954)
Red Blood Cell with continuous viscoelastic protein
network lining membrane.
Advantages: suspended in solution or attached to a
surface
Also for viscoelasticity of leukocytes and
monocytes(Chien 1984, Dong 1988, Richelme 2000)
Micropipette experiment of cells




Model: liquid drop with cortical tension
Measured quantities: cortical tension in membrane,
cytoplasmic viscosity, cell elasticity
for leukocyte under some conditions
Elastic body model (f~2.1)
2 L p
P 
E f
3 Rp

(2.6)
Liquid-like flow of cells, viscosity

R p p
Rp
 d Lp 

 m (1 - )
R
 dt 
(2.7), m  9
Shearing & Compression
L





Attach cells at top and bottom to glass surface moved w.r.t.
each other in compression, extension, or shear.
Fibroblasts adhered to glass surface coated with adhesion
proteins (fibronectin) on rigid plate
Second flexible plate on top surface
PZT motor drives rigid plate to impose strain of cell and
measure deflection of flexible plate to calculate stress.
Imaging internal structure at same time
Fluid Flow
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Cell like vascular endothelial or osteocyte within
bone matrix are regularly exposed to fluid stresses
Cell senses shear stress, crucial for regulatory
processes
In vascular endothelial, mechanosensing control
production of protective ECM
In bone, mechansensing is basis of bone repair and
adaptive structure processes
In vitro shearing of osteocytes in parallel plate flow
chamber
Fluid flow system to stimulate
mechanosensitive bone cells
• Monolayer of
osteocytes
• Controlled shear stress
• NO as func. of flow
rate
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Strain filed within cell measured by endogeneous
fluorescent vimentin (Helmke 2001, 2003)
Inhomogeneous strain field, focused on localized area
Sites of mechanosensing
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Large strain location if distortion of sensing element is required
Focused stress, small strain but high force
Numerical simulations applied to cell & fluid passing over
it.
Combined FEM & CFD to model flow across surface of
adhering cell & calculate shear stress in different spots.
Without knowing intracellular material inhomogeneities
(linear elastic & isotropic)
Also applied to cells manipulated by AFM, magnetic beads,
substrate stretching, useful methods!
z
Optical traps (Laser Tweezers)
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Laser beam forcused through high numerical aperture
r
objective
lens to 3D trap refractive particles (silica or
latex beads)
Force acting on bead: approximation for small particles
(Gaussian focus, size < laser wavelength)
Gradient forces
a  a 'ia " complex polarity
I 0 : laser intensity w0 : beam radius at focus
2
Fgr  -a ' I 0 r
2
2
0
w
e
w4 ( z )
-
w (z)
(2.8)
z0 
 w02
2
: Rayleigh range, k m 
: wave vector
m
m
m : wave length in medium with refractive index n m
2r 2
2
w(z)  w 0
2r 2
w  1
2r  - w 2 ( z )
F  -a ' I 0 z

e
z  w 4 ( z ) w6 ( z ) 
a
g
4
0
2
0
 z
1    : beam radius near focus
 z0 
(2.9)
2 0 0: vacuum wavelength of light
NA 
 D D: diameter of beam waist
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Scattering forces




2r
2
2 
2
2


- 2

w
w0
r z - z0 
 k m 1  e w (z)
Fsa  a " I 0 2
2
2
w ( z )  
2 z 2  z02  z0 w ( z ) 


2
0
Fsr  a " I 0
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2
0
w km r
e
2
w ( z ) R( z )
-
2r 2
w2 ( z )
2
(2.10)
(2.11)
Stable trapping: gradient force wins over the trapping
force; Stability depends on geometry and properties of
trapped particle & surrounding medium.
Particle size & relative index of refraction n=np/nm
Scattering force ~ d6, gradient force ~ d3
(A) Origin of Fscat and Fgrad for high index sphere displaced from TEM00 beam axis.
Ashkin A PNAS 1997;94:4853-4860
©1997 by The National Academy of Sciences of the USA
(A) Geometry of levitation trap.
Ashkin A PNAS 1997;94:4853-4860
©1997 by The National Academy of Sciences of the USA
Working principle
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Trappable bead attached to surface of cell & deform
cell locally;
Advantages: no mechanical access, bead of um size
yield high resolution,
Disadvantages: force < 100pN << micropipetts, AFM;
local heating;
Using interferometric method, great accuracy force &
displacement (sub nm, sub pN)
Fast detection: 10us useful for cell viscoelasticity
Possible to differentiate active, motor driven response
& passive viscoelasticity!
Examples
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Human RBC, 2D shear modulus 2.5~200 uN/m
Magnetic Methods
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Magnetic fields to apply forces and/or torques to
ferromagnetic or paramagnetic particles adhered
to cells. (large force, no open surface)
Difficult to establish homogeneous field
gradients; limited video rates.
Freundlich & Seifriz (1922), insert nickel or
magnetite particle to large cell, magnetic field
gradient generated by electromagnet, impose
force on bead whose displacement measured by
microscope; force calibrated by using fluid with
known viscosity.
Magnetic manipulation system
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Crick & Hughes (1950) modification:
Magnetize particle with large magnetic field & use a
smaller probing magnetic field at different angle to twist
particles.
Phagocytic cells engulf magnetic particles to avoid
damage to cell
Motion of embedded beads depends on probe size;
bead might interact with & stick to cytoskeleton cause
active motion!
How beads are coupled to the network ?
Summary
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Experimental methods for mechanical
properties of cells
Macrorheology, diffusion, sedimentation
Mechanical indentation
Mechanical tension
Shearing & compression
Optical traps
Magnetic methods
3. Continuum elastic or viscoelastic
models for the cell
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Continuum model: when smallest length scale is
larger than dimensions of microstructure
Whole-cell deformation: 1~2 order
Micropipette aspiration of erythrocytes or
neutrophils
Magnetocytometry: bead size & deformation
>> mesh size of cytoskeleton network
Continuum model: no constraints on isotropy,
inhomogeneity of properties
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Constitutive law - stress-strain relation; linear
elastic to viscoelastic
a coarse-grain approach replacing contribution of
cytoskeleton’s stress discrete fibers with averaged
constitutive law
Review of elastic and viscoelastic continuum multicompartment description of the cell
Finite-element-based 2D & 3D models of the cell
with compartments (membrane, actin cortex,
cytoskeleton & nucleus)
Contrasting computational results against
experimental data obtained from different methods.
Purpose of continuum models
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Analyze experiments probing single cell
mechanics
Evaluate force level sensed by various parts of
cell in vitro or in vivo
Compare theoretical and computational
predictions against experimental observations
and deduce mechanical properties of cell
Examples: magnetocytometry, micropipette
aspiration, microindentation, AFM
RBS Aspiration
Neutrophil indentation (Left) & passage (right)
Effect of fluid shear
AFM indentation
Simulation of optical tweezers test
on RBC
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Comparison of continuum models with
corresponding experiments can help to
distinguish active biological responses of the cell
from passive mechanical deformation.
Evaluate strain & stress under biological
conditions: endothelium, neutrophils,
fibroblasts…
Principles of continuum models
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Continuum model provides displacement, strain
and stress fields induced in cell, given initial
geometry, mechanical properties, and boundary
conditions (displacements or force on cell
surface)
Laws of continuum mechanics are used to
solved for distribution of stress and
deformation
Discretize cell volume into small elements using
FEM techniques.
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Assume body and inertial force are negligible
Conservation of linear momentum
σ  0
where s is the Cauchy stress tensor
Boundary conditions
Traction boundary (surface force)
 Displacement boundary (displacement =0)
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Material mechanical
properties
Cytoskeleton: difficult to model,
active/passive,
 Nucleus: stiffer & more viscous, elastic,
18Pa~10kPa
 Cellular membrane: thin (5nm), viscoelastic
time constant 10us
 Cortex: stiffer than rest of cytoskeleton,
RBC, cortical tension
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Examples of studied cells
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Blood cells: leukocytes & erythrocytes
Adherent cells: fibroblasts, epithelial cells &
endothelial cells
Geometry of model
Deformed shapes
FE model of cell monolayer pulled
by magnetic cytometry
FEM simulation of cell contact sites
on basal cell surface (focal adhesion)
Limitations of continuum model
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Aim at passive dynamics of cells
Not account for active biology: deformation and
stresses experienced as a consequence of
biochemical response of cell to mechanical
loading.
May isolate phenomena involving active biology
from passive mechanical response of the cell
Continuum model might simulate active processes
through time-dependent properties or residual
strains linked to biological processes
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Lack of description of cytoskeletal fibers, not
applicable for micro-manipulation of cell with a
probe <= cytoskeletal mesh (0.1-1um)
Exclude Brownian motions due to thermal
fluctuations of cytoskeleton, network node &
cell motility
Limited number of time constants to
characterize cell’s viscoelastic behavior
Need continuous spectrum of time scales
(QLV?)
Conclusion
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Continuum mechanical models useful in exploiting &
interpreting experimental results from probing single
cell or cell monolayer
Identify stress & strain patterns induced within the cell
or mechanical properties of cell compartments.
Predict forces experienced within cells in vivo and form
hypotheses on how cell might sense & transduce forces
into shape change or gene expression
Viscous and viscoelastic models is needed